Simplify and expand the following expression: $ \dfrac{2}{5y - 35}- \dfrac{1}{y - 6}+ \dfrac{4}{y^2 - 13y + 42} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{2}{5y - 35} = \dfrac{2}{5(y - 7)}$ We can factor the quadratic in the third term: $ \dfrac{4}{y^2 - 13y + 42} = \dfrac{4}{(y - 7)(y - 6)}$ Now we have: $ \dfrac{2}{5(y - 7)}- \dfrac{1}{y - 6}+ \dfrac{4}{(y - 7)(y - 6)} $ The least common multiple of the denominators is: $ 5(y - 7)(y - 6)$ In order to get the first term over $5(y - 7)(y - 6)$ , multiply by $\dfrac{y - 6}{y - 6}$ $ \dfrac{2}{5(y - 7)} \times \dfrac{y - 6}{y - 6} = \dfrac{2(y - 6)}{5(y - 7)(y - 6)} $ In order to get the second term over $5(y - 7)(y - 6)$ , multiply by $\dfrac{5(y - 7)}{5(y - 7)}$ $ \dfrac{1}{y - 6} \times \dfrac{5(y - 7)}{5(y - 7)} = \dfrac{5(y - 7)}{5(y - 7)(y - 6)} $ In order to get the third term over $5(y - 7)(y - 6)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{4}{(y - 7)(y - 6)} \times \dfrac{5}{5} = \dfrac{20}{5(y - 7)(y - 6)} $ Now we have: $ \dfrac{2(y - 6)}{5(y - 7)(y - 6)} - \dfrac{5(y - 7)}{5(y - 7)(y - 6)} + \dfrac{20}{5(y - 7)(y - 6)} $ $ = \dfrac{ 2(y - 6) - 5(y - 7) + 20} {5(y - 7)(y - 6)} $ Expand: $ = \dfrac{2y - 12 - 5y + 35 + 20}{5y^2 - 65y + 210} $ $ = \dfrac{-3y + 43}{5y^2 - 65y + 210}$